Determine the remaining Laplacian eigenvalues

Determine the remaining eigenvalues of the quotient matrix associated with the Laplacian matrix of the generating graph of Z_n, beyond the eigenvalues already identified and the bounds obtained through Weyl's inequalities.

Background

The Laplacian spectrum is expressed as a collection of explicitly known eigenvalues together with the spectrum of a smaller symmetric quotient matrix L_Q. The authors identify 0, φ(n), and n as eigenvalues of L_Q and derive upper and lower bounds for its remaining eigenvalues using the matrix decomposition L_Q = -\tilde Q_n + D_L. The complete determination of the remaining eigenvalues is left unresolved because the authors find solving for them challenging.